On the self-replicating properties of Riemann zeta zeros: A statistical study

TL;DR

This study investigates the statistical properties of differences between Riemann zeta zeros, revealing that subsets of zeros contain information about lower zeros, with distributions exhibiting specific skewness and variance patterns related to zero locations.

Contribution

It uncovers the self-replicating properties of zeta zeros and characterizes their difference distributions, providing new insights into their statistical structure and information content.

Findings

Subsets of zeros encode information about lower zeros.

Distribution skewness decreases when crossing a zero.

Variance shows local extrema at zeros.

Abstract

We study distributions of differences of unscaled Riemann zeta zeros, $\gamma-\gamma^{'}$, at large distances. We show, that independently of the height, a subset of finite number of successive zeros knows the locations of lower level zeros. The information contained in the subset of zeros is inversely proportional to $ln(\gamma/(2\pi))$, where $\gamma$ is the average zeta of the subset. Because the mean difference of the zeros also decreases as inversely proportional to $ln(\gamma/(2\pi))$, each equally long segment of the line $\Re(z)=1/2$ contains equal amount of information. The distributions of differences are skewed towards the nearest zeta zero, or at least, in the case of very nearby zeros, the skewness always decreases when zeta zero is crossed in increasing direction. We also show that the variance of distributions has local maximum or, at least, a turning point at every zeta zero, i.e., local minimum of the second derivative of the variance. In addition, it seems that the higher the zeros the more compactly the distributions of the differences are located in the skewness-kurtosis -plane. The flexibility of the Johnson distribution allows us to fit the distributions nicely, despite of the values of skewness and kurtosis of the distributions.